the doppler equation in range - nasa · the doppler equation in range and range rate measurement...

i

I 1

i'

I

THE DOPPLER EQUATION IN RANGE AND

.. RANGE-NRATE MEASUREMENT ,

GPO PRICE $ /

CFSTI PRICE(S) I

Hard copy (HC)

Microfiche (MF)

W 653 July 65

i B

- L lNA$A CRlOR TMX OR AD NUMBER)

ITHRU) 1

I (CODE1 I

L

OCTOBER 8,1965 \

GODDARD SPACE FLIGHT CENTER GREENBELT, M A R Y L A N D

.

https://ntrs.nasa.gov/search.jsp?R=19660010159 2018-05-29T16:29:49+00:00Z

X-507-65-385

..

THE DOPPLER EQUATION IN RANGE AND

RANGE RATE MEASUREMENT

October 8 , 1965

Goddard Space Flight Center Greenbelt, Maryland

. . THE DOPPLER EQUATION IN RANGE AND

RANGE RATE MEASUREMENT

SUMMARY

The Doppler equation and its use for range and range rate measurements is discussed, including special relativistic effects, the meaning of integrated Doppler, basic limitations in the electronic measuring equipment and, the dependence between range and range rate measurements.

The emphasis is on the two-way Doppler, which is of particular interest for range and range rate measurements. It is shown that the classical and relativistic two-way Doppler equations are identical. All derivations are based on time, propagation delay and phase rather than on frequency because the electronic equipment is only capable of measuring finite differences in time and phase. This approach also unifies the treatment of both classical and relativistic Doppler and the integration of Doppler frequency.

It is also shown, that range and range rate measurements are not independent measurements and that range rate measurements can be performed much more accurately than range measurements. The contributions of range measurements to orbit determination a re therefore rather limited except for near earth phases of a mission.

iii

. .

CONTENTS

Page

INTRODUCTION ........................................ 1

1. "CLASSIC" DERIVATION OF THE DOPPLER EQUATION.. . . . . . . . 1

2. RELATIVISTIC DERIVATION OF THE DOPPLER EQUATION.. . . . . 9

3. VELOCITY ABERRATION.. ............................. 17

4. INTEGRATION OF DOPPLER FREQUENCY . . . . . . . . . . . . . . . . . 19

5. LIMITATIONS OF RANGE MEASUREMENTS AND DEPENDENCE BETWEEN RANGE AND RANGE RATE MEASUREMENTS.. . . . . . . 24

V

THE DOPPLER EQUATION IN RANGE AND RANGE RATE MEASUREMENT

INTRODUCTION

Many papers have been written about the Doppler frequency shift of electromagnetic waves and the use of this frequency shift for determination of range and range rate. Then why write another paper? There are several reasons. Not all papers have come to the same conclusions, especially with regard to relativistic effects and the significance of higher order terms in the Doppler equation. It is the objective of this paper to clarify these matters. Another objective is to take into account the limitations in making physical measurements and to interpret the measurements correctly. We cannot measure a frequency instantaneously, a certain time is needed for the measurement. Of particular interest for range and range rate determination is the technique of integrating the Doppler frequency over a finite time. The correct interpretation of the integrated Doppler frequency is therefore as important as the correct derivation of the Doppler equation. It is shown in Chapter 4 that the integrated Doppler frequency is nothing but a measure of propagation delay. This fact is used in Chap- ter 5 to discuss the dependence between range and range rate measurements.

The emphasis in this paper is on time, propagation delay, and phase rather than on frequency. The importance of correct "time keeping" is stressed both in the derivation of the Doppler equation and in evaluating the integrated Doppler frequency. Emphasis is placed on the two-way Doppler as being the most important one for tracking of the Apollo vehicles with the Unified S-Band trackers.

Some of the derivations included in this paper can be found in the literature. Because of the tutorial nature of this paper they have been included in order to present a unified treatment of the Doppler frequency and its integration.

1. "CLASSIC" DERIVATION O F THE DOPPLER EQUATION . An electromagnetic wave of angular frequency at is transmitted from trans-

mitter Tr in Figure 1 at time t,. The amplitude is assumed to be time independent and using complex notations, we can write the time dependence of the wave as exp { j y ) where

w =- drp d t ,

1

Transmitter Tr Ground Receiver Re

Figure 1-Two Way Doppler Geornetri

The wave will reach the vehicle Ve in Figure 1 at a later time t , given by

rl t = t, + - C

where r, = range between transmitter Tr and vehicle Ve

c = propagation velocity of the wave.

An observer at the vehicle Ve will observe an angular frequency 0)"

w = - drp d t

(1-3)

The ratio between uV and at is easily obtained from Equations (1-1) and (1-3)

w d t ,

wt d t - - - -

2

We now choose a frame of reference, in which the transmitter Tr is stationary. In this frame rl is a function of vehicle position only and as the vehicle Ve receives the electromagnetic wave at time t , rl has to be considered to be a function ofwt. With this in mind, differentiation of Equation (1-2) with respect to t yields

tl 1 =1 - = I - - - dt c d t

and thus

(1-5)

which is the one-way Doppler equation for a moving observer.

The wave wv is now reflected or instantaneously retransmitted from the vehicle Ve to a ground receiver Re. It is assumed that the receiver Re is stationary relative to the transmitter Tr. The range r, (see Figure 1) is therefore a function of the time of retransmission t . The retransmitted wave will be received by the ground receiver Re at a time t, given by

rl + r2 t , t -

t , = t + - = r2

C C

and the received angular frequency wr is

da,

and thus u d t ,

w t d t 2 r - - _ -

(1-7)

(1-9)

Differentiation of Equation (1-7) yields, observing that rl and r, are functions of t

(1-10) dt2 . 1 dr2

I + - - c d t

3

and with Equation (1-9) we obtain

, 1 dr1

1 + - - 1 dr2 c d t

The angular Doppler frequency wd is defined as

w = 0, - at d

and using Equation (1-11)

1 drl 1 d'2 -- t-- c d t c d t .w

c d t

w - t

1 dr2 d - -

1 + - -

(1-11)

(1-12)

(1-13)

which is the two-way Doppler equation.

Note that all derivatives in the Doppler equation reflect the vehicle time t rather than the time t2 at the ground receiver.

Equation (1-13) may also be expressed as a series.

drl dr2 a,, = - {: (2 t?) - f ( dt2 t (2r) t . . 1 wt (1-13a)

Although the series expansion is in common use we prefer to use the exact form, as given by Equation (I-13), in this paper.

Another useful form of Equation (1-13) is obtained by solving for the range rate

d

t

w

w d t (1-13b)

4

Of special interest is the case where the transmitting antenna coincides with the receiving antenna. In this case rl = r2 and the Doppler equation can be simplified to

and

?

w d = - 2 c d t .iL) . 1 d r I + - -

c d t

1 d r 1 d2r - - - - - + . . . C d t c2 dt2

*d C -

dr - *t - -- wd d t

2 +-

(1-14)

(1-14a)

(1-14b)

The two way Doppler equation can easily be generalized. In some applications the vehicle transponder multiplies the received frequency by a factor k before retransmission. The retransmitted angular frequency is then kwv and Equation (1-11) changes to

The angular Doppler frequency for this case is defined as

ud = o - k w t

5

and thus 1 dr1 1 d'2

1 dr2

-- t - - . k a t c d t c d t

1 + - - a - d - -

c dt

(1-15)

So far we have assumed that the propagation velocity is constant and that the propagation path is a straight line. The Doppler equation can be derived under much more general conditions. Let the delay be 71 for the propagation from the transmitter T r to the vehicle V e and 72 fromVe to the receiver Re. Equation (1-7) can then be written, see Figure 2.

t = t t r 2 = t l t T 1 t r 2 (1-16) 2

where 7, and 72 a r e functions of t. Equation (1-9)

w dt , r - - _ -

at dt2

ti

Vehicle Ve time t = t l + T ~

me t l

Transmitter Tr Ground Receiver Re

Figure 2-Propagation Delays

6

(1-9)

time tp = t + T ~

holds true for arbitrary transmission media. Differentiating (1-16) we obtain

Combining Equation (1-9) and (1-17) with (1-12) results in

dT1 dT2 +- -

d t d t t

Wd = - w 1 1 d72 1 + - -

‘ c d t

which is the two-way Doppler equation for a general propagation media.

(1-17)

(1-18)

This equation clearly shows that the Doppler frequency is caused by the time derivatives of the propagation delay and is independent of whether the propagation delay varies because of variation in range o r variation in the propagation media.

For the special case

rl r2 l c = - r 2 = - C

Equation (1-18) reduces to (1-13). The problem in the general case is, of course, to establish what functions T~ and r2 are of t. Other generalizations such as transponder frequency multiplication and transponder time delay are easily in- corporated into Equation (1-18).

So fa r , the Doppler equations have been expressed in space vehicle time t and the equations therefore reflect the range rate at time t. If we want to know the actual range rate at observation time t 2 , a correction has to be ap2lied. The range has to be expressed as a function of t2 . From

7

(1-19)

we obtain

. .

and after expansion into a Taylor series

In taking the derivative and observing that t, is a function of t given by Equati;.n (1-19) we find

With the aid of Equation (1-20) we can eliminate r,(t), resulting in

(1-21)

(1-22)

In the same way we obtain

r 2 ( t 2 ) d r 2 ( t 2 ) d 2 r , ( t 2 ) +- r 2 ( t 2 ) d3r , ( t , ) } + . . . (1 -23) ' - C2 { d t 2 (dt,), (dt,I3

8

Substitution of Equations (1-22) and (1-23) into the Doppler equations will yield equations which express the Doppler shift as a function af range rate at observation time t, . In particular, the two-way Doppler Equation (1-13) will transform into

dr2 + (dr2I2 d2rl t -k r2

C (d t2 )2

drld2r2 +2dr,d2rl +3dr2d2r, d r 2 2 drl t d r , ‘(q) d t , t r: (dt,)3

(1 -24)

where rl and r, now are functions of t,. For the special case of the coinciding receiver and transmitter antenna this equation reduces to

(1-25)

It is obvious from the preceeding that the Doppler equations expressed in observation time t, are rather lengthy. Also, they are no longer exact and in the above derivation terms containing l/c3 and higher orders have been neglected. It might be simpler in many applications to use the Doppler equations for time t and to use a time correction as given by Equation (1-19).

2. RELATIVISTIC DERIVATION OF THE DOPPLER EQUATION

Reviewing the classical derivation of the Doppler equations we find that the following assumptions were made:

a. The propagation velocity c was assumed to be constant both for transmission from the ground and from the vehicle.

b. The reflection or retransmission from the vehicle is instantaneous. t

c. Transmission and reception occurs in the same reference frame.

9

It therefore seems very likely that no relativistic effects should occur in the two-way Doppler. A relativistic derivation is given here in order to verify this assumption. The derivation is based on the Lorentz transformation and thus takes the effects of special relativity into account.

We choose two reference frames S and S' so that the transmitter Tr and the receiver Re are stationary in the S frame and the vehicle Ve is stationary in the S' frame. Without loss of generality we can orient the frames so that their X - and y-axes are in the same plane and their x-axes are parallel to the velocity vector V of the vehicle. In addition to this, we let the x-axes coincide as shown in Figure 3.

Y ' Y

s

Tr (transmitter)

Ve (vehicle)

a' 7P=)-v / S'

Figure 3-Reference Frames for the Lorentz Transformation. Transmission from the Ground Transmitter Tr.

From the S frame as electromagnetic wave which we, as in Chapter 1, write e x p { j 'p) is transmitted at time t, and reaches the vehicle at time t , given by Equation (1-2)

rl t = t + -

1 c

From Figure 3 we see that

and hence

x cos a t y s i n a t = t , + C

By means of the Lorentz transformation x, y and t from the S frame can be transformed into X I , y' and t' in the S' frame, see reference [l I

V t ' +- X I

C2 t = (2-3)

By applying Equation (2-3) to Equation (2-2) we can solve for the time t, of transmission (from the S frame) in terms of time t ' of reception (in the S' frame)

V t' +- X'

CZ x' -t v t ' t = c o s rr + y' s i n a 1

- C2

The transmitted angular frequency is

J

CZ t = c o s rr + y' s i n a 1 (2-4)

. w =- dT dt,

11

and the angular freqiency received by the vehicle is

and thus Q d t ,

ut d t ' v - - _ -

From Equation (2-4) we obtain

I -L c o s a w v - dt , C

But

rl

dt v cos a = -

and thus

1 - - - V c d t - - _

. - (2-5)

(2-7)

Comparing Equation (2-7) with Equation (1-6) we see that there is a difference between the classic and relativistic one-way Doppler equation. Expapding Equa - tion (2-7) into a series we find that the relative difference is 3 2 plus higher order terms.

The direction cosine in the S' frame can be found from Equation (2-4) which reads after rearranging terms

V

C 1 - - - - s a

t = 1 1 - _ .t ' C

. -

c o s a - - V )-$si.- )} Y' C

x ' t V V

C C 1 - - c o s a - - cos a

,

12

and the direction cosine in the S' frame is

V cos a - -

C cosa' = V

C P ----cos a

(2-9)

Let us now retransmit the electromagnetic wave from the moving frame S' to a receiver R e in the stationary frame S as shown in Figure 4. The wave is transmitted from the vehicle at time t' and reaches the receiver R e at time t i as seen from the S' frame.

(2-10) 1 r; t# = t + - C 2

Y

f

S s'

+ V Ve (vehicle)

L - t X '

X Re (receiver) v Figure 4-Reference Frames for the Lorentz Transformation Reception by the Ground Receiver Re

13

The time t, of reception in the s frame is again obtained by means of the Lorentz transformation. Observing that the sign of the velocity has changed, the Lorentz transformation reads

Y' = Y

z ' = z = O

V

t , -- x t ' = C2 2

With, see Figure 4,

r; = x' c o s (n t p ' ) + y' s i n (n + p ' )

we obtain from Equations (2-10) and (2-11)

V

C 1 - - c o s p'

t ' = 1 - - i; C

( - c o s . . + - V

C

p - f cos p' x -

JI - s i n

V

C 1 - - c o s p'

(2-11)

(2-12)

(2-13)

14

and the received angular frequency wr is thus,

and the direction cosine in the S frame is

V

C c o s p' - -

1 - - c o s p' c o s p =

V

C

o r

V

C cos p + -

1 + - c o s p

c o s p' = V

C

(2-14)

(2-15)

(2-16)

Equation (2-14) contains the direction cosine of the S' frame. Substituting c o s ,/3' from Equation (2-13) yields

(2-17) w V

C v 1 + - c o s p

15

The two-way Doppler equation is obtained by combining Equations (2-6) and (2-17)

V 1 -- cos a

t 1 + - c o s p

wr - C - _ w V

C

o r , 1 d r l

t . 1 d r , w 1 + - -

c d t

since

v c o s p d'2 - d t

(2-18)

(2-19)

(2-20)

Equation (2-18) is identical to the "classic" two-way Doppler equation and contains no "relativistic terms.'' Note that Equation (2-18) contains only the direction cosine of the S frame. This is a logical choice if we want to exliress the two-way Doppler equation in terms of range rate referenced to the S frame, see Equation (2-19). We could also use the direction cosine in the S' frame

V

r C 1 - - c o s p'

c4)

(2-21)

without encountering relativistic terms . I1 However, if direction cosines a re mixed both from the S and S' frame

r C C ) (1 - z c o s .) (1 - - c o s p' V w

(2-22)

n

C'

16

then the Doppler equation will contain "relativistic terms," but there seems to be no reason for using the Doppler equation in the "mixed" form.

3. VELOCITY ABERRATION

An electromagnetic wave, reflected in a corner reflector on the vehicle, will not be reflected back to the stationary reference frame at the same angle as it was transmitted. Both a non-relativistic and a relativistic derivation of this aberration is given here.

A. Non-Relativistic Derivation

The velocity components of the wave in the stationary reference frame are c s i n a and c c o s a as shown in Figure 5. In the moving reference frame the velocity components a re c s i n a and c c o s a - v . After reflection the absolute values of the components are I C s i n QI and I C C O S a - 2 v 1 . Thus

o r

V 2 - . s i n a C tan 24=

2v 1 - c cos a

where 2 4 is the aberation angle.

B. Relativistic Derivation

(3-1)

The relativistic aberation can be obtained from Equations (2-9) and (2-15) For the corner reflector we have Q' = p'. With p = a -+ 2 4 we thus obtain

V V

C =os ( a t 24) t - C O S a - - = - -

V V l - - c o s a C l + - c o s ( a + 2 ~ ) C

17

Y Y I c sin a

c sin a

+ V

Figure 5-Non-relotivistic Velocity Aberration

Solving for 24 yields

V V2

C C2 2 - s i n a - - s i n 2 a

t a n 24 = V V2

C C2

1 - 2 - c o s a + - c o s 2 a

o r

V

C - s i n a

t a n d =

(3-2)

(3 -2a) V

C 1 - - c o s a

The difference in non-relativistic and relativistic aberration is best seen if we expand Equations (3-1) and (3-2) into series.

Non-Relativistic :

(3-3)

18

Relativistic :

t a n 24 = - 2v s i n a ( l . r c 0 s a t - t V2

C2 C

(3-4)

The difference between the two derivations is in the second order term and the relative difference is therefore of the magnitude (v/c )2 .

If we instead require that the angles of transmission and reception at the ground are identical, i.e., a = p, then the angle of transmission p' from the vehicle has to offset from the angle of reception a' at the vehicle. With the offset angle 2$

2 4 + a' = p' (3-5)

we obtain from Equations (2-9) and (2-15)

2 - s i n V

1 + 2 - c o s V

a' + - v2 s i n 2a' C C 2

tan 2 $ = - a' + V2 - c o s 2a'

C C 2

o r V - s i n a'

t an $I = - V

C 1 .t - c o s a'

(3-6)

(3-7)

4. INTEGRATION OF DOPPLER FREQUENCY

In the previous Chapters we derived the Doppler frequency as a function of the range rate of a moving vehicle. In performing range and range rate measurements we are faced with a different problem: If we measure the Doppler frequency what can we say about the range and range rate?

Frequency cannot be measured instantaneously. The most commonly used methods for measuring the Doppler frequency is to count the number of Doppler cycles during a fixed time period or to measure the time required for the reception of a fixed number of Doppler cycles. Both methods imply an integration of Doppler frequency and the integrated Doppler frequency will be discussed in this Chapter.

19

Let u s integrate the two-way angular Doppler frequency at the receiver during a time interval T, . We obtain

by using Equations (1-9) and (1-12). But from Equation (1-18)

t , = t, t 7, - T2

and thus

In Figure 6 @ and @ denote the vehicle positions corresponding to the beginning and the end of the integration interval T,. With the notations in Fig- ure 6, Equation (4-2) may also be written

Equation (4-3) shows clearly that the integrated Doppler frequency shift is a measurement of the difference in propagation delay when the wave travels the path T r -0 - R e and the path T r -+ @ - Re. It is seen from Figure 6 that the vehicle is in position @ at vehicle time t

t = t , - r 2 1 (4 -4 1 4

and in position @ at vehicle time t + T

(4-5)

20

Ve travels from a t o @ i n the

x

Integration in terva I : \ \ t 2 t o t 2 + T 2

Tr (transmitter) Re (receiver)

Figure 6-Integration of Doppler Frequency

The above equations a re general to the extent that no assumptions have been made about the index of refraction of the propagation media.

If we want to interpret the measured difference in propagation delay as a difference in range, then assumptions have to be made about the index of refraction. Let us assume that the propagation media is vacuum. Then

- ‘12 etc. - ‘1 1 rll - 7 ’ r12 -c

21

and

w t

t 2+T2

=2

ud d t , -5 -- {(r , , t r,,) - ( r l l t r,,)) I C

where

r l l = ‘11

r21 = r 2 i’ a t

r21 2 - 7

t = t

With the following use full notations

r12 - r l l = Ar,

= Ar, r 2 2 - r 2 1

t2+T2 wd d t , = A + = i n t e g r a t e d D o p p l e r f r e q u e n c y I t 2

Equation (4-6) can be written

c A 4 Ar, t Ar, = -- w

t

(4-6)

. .

(4-7)

In other words, the integrated Doppler frequency A4 is proportional to a change in range.

The time T required for the vehicle to travel from @ to @ is according to Equations (4-4) and (4-5)

T = T, -- C

22

(4-8)

If we define the average range rate as

then Equation (4-7) yields

(4-10)

At the ground receiver R e only the time interval T2 is known and not the time interval T. From Equations (4-8) and (4-9) we obtain

T2 T =- ‘a2

1 t- C

and thus

(4-11)

For the case of the coinciding ground transmitter and receiver antenna, Le. A r , = A r 2 , Equation (4-11) reduces to

(4-12)

23

Equation (4-11) is the integrated two-way Doppler equation. Observe the formal similarity between the integrated Equation (4-1 1) and the non-integrated Equation (1-13b). E we define the average angular Doppler frequency wda

we see that Equation (4-11) contains only averaged quantities o r finite differences. Equation (1-13b) 03 the other hand contains only differentials and has therefore an “instantaneous” character. Although the Equations (4-11) and (1-13b) are formally similar, they a re only identical in the limit when T2 -. 0. The difference between the average range rate ia and the instantaneous range rate considerable, see reference (3).

may be

In order to associate a time with the middle of the integration interval 1 t + - T , expressed in vehicle time, is chosen. This choice is somewhat arbi- 2

trary but has the advantage of minimizing the difference between ia and i, see reference (3). With the aid of Equations (4-4) and (4-5) this time can be expressed in ground receiver time

1 T2 1 t + -T = t + - -- 2 2 2 2 ( 7 2 1 +722’

o r

1 T2 1 (r21 f r22’ t + - T = t2 + - - -

2 2 2c

(4-13)

(4-13a)

where wave propagation through vacuum is assumed in the last equation.

5. LIMITATIONS O F RANGE MEASUREMENTS AND DEPENDENCE BETWEEN RANGE AND RANGE RATE MEASUREMENTS

Range measurements consist of the measurement of the two-way propagation delay of an electromagnetic wave. The same antenna is used both for transmission

24

and reception. In order to compare range and range rate measurements we therefore consider the two-way Doppler equations for the special case of coinciding transmitting and receiving antennas. The Equation (4-3) for the integrated Doppler frequency reduces for this case to

with notations as shown in Figure 7. From this equation we can draw a number of interesting conclusions :

1. The integrated two-way Doppler frequency is proportional to the difference between the two-way propagation delays at the beginning and at the end of the integration interval. A range measurement consists of the measurement of the two-way propagation delay. The integrated two-way Doppler frequency is thus proportional to the difference of two range measurements, namely the measurements of r2 and rl as shown in Figure 7.

Ve (vehicle)

Figure 7-Compari son of Integrated Doppler Frequency and Range Measurement

25

2. Only the propagation properties along r l and r z are of importance and these propagation properties are the same for the integrated Doppler frequency and for range measurements of r l and r2. Note that the propagation properties of the propagation medium between r l and r 2 are of no importance.

3. If we write the range in the form

we can consider ro as an integration constant. The integrated Doppler frequency, i.e., a range rate measurement, will not give us any information about ro. A series of range measurements therefore contains more information than the corresponding range rate measurements.

A very important consequence of the above conclusions is: Range and range rate measurements a re not indeDendent measurements as far as Dropaaation properties are concerned. Both a r e measurements of time delay and a range rate measurement is identical to the difference between two range measurements. Measuring errors introduced by the ground equipment are , on the other hand, to a certain extent independent. An example is the oscillator drift, which will effect the integrated Doppler frequency o r range rate during the hole integration interval but a range measurement only during the propagation time T. For a complete analysis of these e r rors the particular measuring scheme employed has to be known and no general statements can therefore be made.

The above conclusion 3 states that a series of range measurements contains more information than the corresponding range rate measurements. If this is so, why a re range rate measurements used? The answer is that range rate measurements can be performed much more precisely than range measurements. In order to resolve the ambiguities in range measurements, the electromagnetic wave (carrier) has to be modulated (side tones, pseudo random codes). The bandwidth allocated for this modulation determines the capability to resolve the fine ambiguities. The range accuracy is limited by the fine ambiguity resolution and is thus bandwidth limited. Most of the information is therefore obtained from the integrated range-rate o r Doppler frequency and the range measurements a re only used for determination of the integration constant rO. As long as the integration process is not interrupted, only one range measurement is required for the determination of ro. However, several range measurements may be used in order to reduce the error in ro by statistical methods.

26

Another interesting question of great practical importance is: What is the contribution of range measurements to orbit determination if both range and range rate measurements are available? From the above, one might suspect that the contribution is rather limited and this suspicion is verified by recent e r ror analysis studies and actual experience. In reference (4) it has been shown by means of e r ror analysis that range rate and angle measurements are adequate for orbit determination of the translunar, lunar and transearth phases of the Apollo missions. From reference (5) (table 1, page 7), we learn that only range rate and angular measurements were used for the Ranger VI1 orbit determination. Except for near earth phases of missions, the contribution of range measurements to orbit determination is therefore rather limited.

A rather high price has to be payed in bandwidth and hardware for obtaining range measurements. It should therefore be seriously considered if range measurements a re really required, especially for deep space and galactic missions where bandwidth and weight are at an extra premium.

27

REFERENCES

1. George Joos: Theoretical Physics (Book) G. E. Stechert & Co., New York o r Walter Weitzel: Lehrbuch der theoretischen Physik (Book) Springer- Verlag, 1963.

2. H. Lass and C. B. Sol1oway:Technical Memorandum 312-94, Jet Propulsion Laboratory, March 22, 1961.

3. "The Range Rate Error Due to the Averaging Techniques of Doppler Measure- ments," by B. Kruger, Report X-513-65-100, Goddard Space Flight Center, March 5, 1965.

4. MSC-GSFC: ANWG Technical Report No. 65-AN-2.0, "Apollo Navigation- Ground and Onboard Capabilities," October 8, 1965.

5. J P L Technical Report No. 32-694, "Ranger VI1 Flight Path and Its Deter- mination from TrackingData,"December 15, 1964.

29

the doppler equation in range - nasa · the doppler equation in range and range rate measurement...

Documents